Figure 1: Irradiance cross-section of beam after propagating 1500 m.

Interest in high-data-rate free-space optical (FSO) laser communication systems has grown significantly in recent years because of the advantages offered by FSO systems over radio frequency (RF) systems. Most advantages are simple consequences of the short wavelengths associated with optical waves. However, there are some drawbacks that arise from the shorter wavelengths used in FSO systems. Among the advantages and disadvantages are the following:
Atmospheric factors are the most serious drawback to FSO systems. Optical turbulence resulting from small temperature variations gives rise to power losses from spreading of the beam beyond that due to diffraction alone, and to temporal and spatial fluctuations of the laser beam known as scintillation. In addition, we may have to account for beam-wander-induced scintillation in a focused beam on a horizontal path or in a collimated beam on an uplink path to space.
In Fig. 1 we illustrate what the irradiance cross-section of a laser beam may look like after propagating 1500 m through extended atmospheric turbulence along a horizontal path close to the ground. The dark spots correspond to a potential fade that may exist for a receiver placed in this position.

An optical wave propagating through atmospheric turbulence will experience irradiance fluctuations (called scintillation) due to small index of refraction fluctuations (i.e., optical turbulence). Theoretical and experimental studies of irradiance fluctuations generally center around the scintillation index

where the quantity I denotes irradiance (intensity) of the optical wave and the angle brackets < > denote an ensemble average, or equivalently, a long-time-average. For constant values of the refractive-index structure parameter, it is known that the scintillation index increases with increasing path length until it reaches a maximum value greater than unity in the regime characterized by random focusing (see Fig. 2). With increasing path length, the focusing effect is weakened by multiple scattering and the fluctuations slowly begin to decrease, saturating at a level for which the scintillation index approaches unity from above. Saturation occurs because multiple scattering causes the optical wave to become increasingly less coherent as it propagates. The theory that we use in this paper is based on recently developed theory contained in Refs. [1] and [2]. Comparisons of this theory with outdoor experimental data and simulation results justify its use under all conditions of optical turbulence.

Figure 2: Scintillation index.

Figure 3: Direct detection system.

Optical receivers are broadly divided into two basic typesXdirect (or power) detecting receivers and coherent receivers. The simplest type of optical receiver for implementation is a power detecting receiver (see Fig. 3). The optical field is always photodetected in the presence of various noise sources, e.g., background radiation, detector noise, and circuit and electronic thermal noise. Increasing the collecting lens aperture diameter beyond the irradiance correlation width of the received optical wave not only increases the average signal level, but decreases the irradiance fluctuation level through a process called aperture averaging.

Figure 4: Probability of fade curves.

An important parameter used to characterize system performance is the receiver signal-to-noise ratio (SNR). False alarm and fade probability both involve the notion of SNR. The reliability of an FSO system operating in the atmosphere can be deduced from the probability density function (PDF) of the irradiance signal. Two models commonly used for this purpose are the lognormal PDF and the recently developed gamma-gamma PDF. However, the lognormal model is limited to weak fluctuations, and even then it generally predicts overly optimistic results.
In Fig. 4 we show the probability of fade versus the mean SNR for the case of a spherical wave with specified Rytov variance . In this case we have included noise in the receiver and used conditional statistics. Also, is a normalized aperture diameter and the false alarm rate (FAR) per bandwidth is specified at .
In digital transmission the desired message is converted to binary symbols (bits) and transmitted as a modulated optical field. The performance measure in such systems is commonly provided by the probability of error, also called the bit error rate (BER). The most basic form of pulsed modulation in binary direct detection receivers is on-off keying (OOK).

Figure 5: Mean BER for on-off keying (OOK).

In the presence of optical turbulence, the probability of error is considered a conditional probability that must be averaged over the PDF of the random signal to determine the unconditional BER. We show the mean BER in Fig. 5 as a function of mean SNR for the case of a spherical wave under various conditions. The case corresponds to weak irradiance fluctuations and we have taken the receiver normalized aperture diameter .For wavelength m and , this corresponds to a path length just over 300 m and receiver aperture = 3.2 cm. For the moderate irradiance fluctuation case , we plot the probability of fade for two normalized aperture sizes corresponding to and . Under the conditions specified above, the path length would be roughly 1 km and aperture diameters and 40 cm, respectively.

Figure 6: Scintillation index of a focused beam on a horizontal path. The dashed curve is based on conventional first-order Rytov theory.

Along a horizontal path, beam wander-induced scintillation may not be a problem for a collimated or divergent beam. However, beam wander effects have to be accounted for if the beam is focused. Recent modeling [3] has shown that beam wander causes an effective pointing error in the scintillation index of an untracked beam that leads to larger values of the scintillation index than that predicted by the Rytov theory. The amount of reduction in the scintillation index when the beam is tracked depends on the tracking method. For example, in Fig. 6 we show recent simulation data (courtesy of G. J. Baker and R. Parenti) of a beam focused at the receiver for both the untracked beam and tilt-removed beam along with theoretical results based on weak fluctuation theory.

Figure 7: Uplink scintillation index.

Figure 8: Probability of fade curves.

For an uplink collimated beam to space we must account for beam wander effects when the beam is untracked. In Fig. 7 we illustrate recent simulation results (courtesy of G. J. Baker) for the on-axis scintillation index as a function of beam radius . Also shown are theoretical curves based on newly developed beam wander theory (solid curve) and conventional Rytov theory (dashed curve). If the beam is tracked, the on-axis scintillation index more closely matches that of the Rytov theory. Lastly, in Fig. 8 we illustrate the effect of beam wander on the fade probability of an uplink collimated beam to a satellite in geostationary orbit. The beam radius at the transmitter is 10 cm and the wavelength is 1.55 microns. Both tracked and untracked beam fade probabilities are illustrated at zenith angles of 0 deg and 45 deg. Detector noise is neglected.

References
[1] L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[2] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[3] L. C. Andrews, et al., “Beam wander effects on the scintillation index of a focused beam,” SPIE 5793 (2005), to appear.